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rad/m
2.915452
rad/m
2
m⁻¹
200
cm⁻¹
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Hz
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m
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rad/m
2.915452
rad/m
2
m⁻¹
200
cm⁻¹
—
Hz
—
m
The Wave Number Calculator computes the wave number $$k$$ — the spatial frequency of a wave — using two methods: $$k = \frac{2\pi}{\lambda}$$ from the wavelength, and $$k = \frac{\omega}{v}$$ from the angular frequency and wave speed. Wave number is a fundamental quantity in physics that describes how rapidly a wave oscillates in space, measured in radians per meter.
In spectroscopy, the term "wave number" often refers to the spectroscopic wave number $$\tilde{\nu} = 1/\lambda$$, measured in cm⁻¹. This convention is standard in infrared spectroscopy and chemistry, where absorption peaks are labeled by their position in cm⁻¹. This calculator provides both the physics convention ($$k$$ in rad/m) and the spectroscopy convention ($$\tilde{\nu}$$ in m⁻¹ and cm⁻¹).
Wave number plays a central role in Fourier analysis, crystallography, quantum mechanics (where $$p = \hbar k$$ relates momentum to wave number), and solid-state physics (Brillouin zones, band structure). It is the spatial counterpart of angular frequency: while $$\omega$$ counts radians per second, $$k$$ counts radians per meter.
The wave number is computed using two equivalent approaches:
From wavelength: $$k = \frac{2\pi}{\lambda}$$
From angular frequency and speed: $$k = \frac{\omega}{v}$$
Both give the same result when $$\omega = 2\pi v / \lambda$$. The spectroscopic wave number uses a different convention:
$$\tilde{\nu} = \frac{1}{\lambda}$$ (in m⁻¹) or equivalently in cm⁻¹ (multiply by 100 when $$\lambda$$ is in meters).
Key relationships include:
A larger wave number means shorter wavelength and more rapid spatial oscillation. In the wave equation $$\psi(x,t) = A\sin(kx - \omega t)$$, the wave number determines how tightly packed the wave crests are. For context, visible light has wave numbers of roughly $$9 \times 10^6$$ to $$1.6 \times 10^7$$ rad/m (or 14,000–25,000 cm⁻¹ spectroscopically).
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A 10 μm infrared photon (CO₂ laser wavelength) has a spectroscopic wave number of 1000 cm⁻¹, a common reference point in infrared spectroscopy.
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A 1 kHz sound wave in air has a wavelength of 34.3 cm and a wave number of about 18.3 rad/m — roughly 3 full wave cycles per meter.
The physics wave number $$k = 2\pi/\lambda$$ is measured in radians per meter and counts full angular cycles per meter. The spectroscopic wave number $$\tilde{\nu} = 1/\lambda$$ counts the number of wavelengths per unit length and is typically expressed in cm⁻¹. They differ by a factor of $$2\pi$$: $$k = 2\pi\tilde{\nu}$$.
Spectroscopic wave number in cm⁻¹ is proportional to energy ($$E = hc\tilde{\nu}$$) and produces conveniently sized numbers for molecular vibrations (400–4000 cm⁻¹). It also avoids the extremely large numbers that result from expressing infrared frequencies in Hz (10¹³–10¹⁴ Hz).
In quantum mechanics, the de Broglie relation connects wave number to particle momentum: $$p = \hbar k$$, where $$\hbar = h/(2\pi)$$ is the reduced Planck constant. A particle's wave number directly encodes its linear momentum, making $$k$$ the natural variable for momentum-space analysis.
The wave vector $$\vec{k}$$ is the three-dimensional generalization of wave number. It points in the direction of wave propagation and has magnitude $$|\vec{k}| = k = 2\pi/\lambda$$. In 3D, the wave equation becomes $$\psi = A\exp(i\vec{k}\cdot\vec{r} - i\omega t)$$.
Visible light spans approximately $$9 \times 10^6$$ to $$1.57 \times 10^7$$ rad/m in physics convention, or about 14,300–25,000 cm⁻¹ in spectroscopic convention. Red light (~700 nm) has the smallest wave number; violet (~400 nm) has the largest.
In crystallography, the Brillouin zone is the region in wave-number (k-space) that contains all unique wave vectors for a periodic lattice. Its boundaries occur at $$k = \pm \pi/a$$, where $$a$$ is the lattice spacing. The electronic band structure of solids is typically plotted within the first Brillouin zone.
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